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## Decomposition of $S_{12}^{\mathrm{new}}(15)$ into irreducible Hecke orbits

magma: S := CuspForms(15,12);
magma: N := Newforms(S);
sage: N = Newforms(15,12,names="a")
Label Dimension Field $q$-expansion of eigenform
15.12.1.a 1 $\Q$ $q$ $\mathstrut-$ $56q^{2}$ $\mathstrut-$ $243q^{3}$ $\mathstrut+$ $1088q^{4}$ $\mathstrut+$ $3125q^{5}$ $\mathstrut+$ $13608q^{6}$ $\mathstrut+$ $27984q^{7}$ $\mathstrut+$ $53760q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
15.12.1.b 2 $\Q(\alpha_{ 2 })$ $q$ $\mathstrut+$ $\alpha_{2} q^{2}$ $\mathstrut+$ $243q^{3}$ $\mathstrut+$ $\bigl(- 22 \alpha_{2}$ $\mathstrut- 560\bigr)q^{4}$ $\mathstrut-$ $3125q^{5}$ $\mathstrut+$ $243 \alpha_{2} q^{6}$ $\mathstrut+$ $\bigl(- 1276 \alpha_{2}$ $\mathstrut- 19468\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 2124 \alpha_{2}$ $\mathstrut- 32736\bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
15.12.1.c 2 $\Q(\alpha_{ 3 })$ $q$ $\mathstrut+$ $\alpha_{3} q^{2}$ $\mathstrut-$ $243q^{3}$ $\mathstrut+$ $\bigl(- 13 \alpha_{3}$ $\mathstrut- 1640\bigr)q^{4}$ $\mathstrut-$ $3125q^{5}$ $\mathstrut-$ $243 \alpha_{3} q^{6}$ $\mathstrut+$ $\bigl(3584 \alpha_{3}$ $\mathstrut+ 27188\bigr)q^{7}$ $\mathstrut+$ $\bigl(- 3519 \alpha_{3}$ $\mathstrut- 5304\bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$
15.12.1.d 3 $\Q(\alpha_{ 4 })$ $q$ $\mathstrut+$ $\alpha_{4} q^{2}$ $\mathstrut+$ $243q^{3}$ $\mathstrut+$ $\bigl(\alpha_{4} ^{2}$ $\mathstrut- 2048\bigr)q^{4}$ $\mathstrut+$ $3125q^{5}$ $\mathstrut+$ $243 \alpha_{4} q^{6}$ $\mathstrut+$ $\bigl(- 12 \alpha_{4} ^{2}$ $\mathstrut+ 460 \alpha_{4}$ $\mathstrut+ 38888\bigr)q^{7}$ $\mathstrut+$ $\bigl(- \alpha_{4} ^{2}$ $\mathstrut+ 1354 \alpha_{4}$ $\mathstrut- 7248\bigr)q^{8}$ $\mathstrut+$ $59049q^{9}$ $\mathstrut+O(q^{10})$

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ $\Q(\sqrt{1609})$ $x ^{2}$ $\mathstrut +\mathstrut 22 x$ $\mathstrut -\mathstrut 1488$
$\Q(\alpha_{ 3 })\cong$ $\Q(\sqrt{1801})$ $x ^{2}$ $\mathstrut +\mathstrut 13 x$ $\mathstrut -\mathstrut 408$
$\Q(\alpha_{ 4 })$ $x ^{3}$ $\mathstrut +\mathstrut x ^{2}$ $\mathstrut -\mathstrut 5450 x$ $\mathstrut +\mathstrut 7248$

## Decomposition of $S_{12}^{\mathrm{old}}(15)$ into lower level spaces

$S_{12}^{\mathrm{old}}(15)$ $\cong$ $\href{ /ModularForm/GL2/Q/holomorphic/5/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 }$